Binary Darboux Transformation Research Articles

Binary Darboux transformation and multi-dark solitons for a higher-order nonlinear Schrödinger equation in the inhomogeneous optical fiber

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schrödinger equation, since dark solitons can be applied in waveguide optics as dynamic switches and junctions or optical logic devices. Based on the Lax pair, the binary Darboux transformation is constructed under certain constraints, thus the multi-dark soliton solutions are presented. Soliton propagation and collision are graphically discussed with the group-velocity dispersion, third- and fourth-order dispersions, which can affect the solitons’ velocities but have no effect on the shapes. Elastic collisions between the two dark solitons and among the three dark solitons are displayed, while the elasticity cannot be influenced by the above three coefficients.

Multisolitons of the $$U(N)$$ generalized Heisenberg magnet model and the Yang–Baxter relation

We use the binary Darboux transformation to obtain exact multisoliton solutions of the $$U(N)$$ generalized Heisenberg magnet model and present the solutions in terms of quasideterminants. In addition, based on using the Poisson bracket algebra, we develop a new canonical approach of the type of the $$r$$ -matrix approach for the generalized Heisenberg magnet model.

Bound-state solitons for a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber

In this paper, we study a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber for two orthogonally polarized optical pulses. With respect to the slowly-varying envelopes of two interacting optical modes and based on the existing binary Darboux transformation, we obtain four types of the bound-state solitons: degenerate-I, degenerate-II, degenerate–non-degenerate, and non-degenerate–non-degenerate bound-state solitons. We graphically analyze the interactions between the degenerate or non-degenerate solitons and four types of the bound-state solitons. When the degenerate solitons interact with the bound-state solitons, amplitudes and widths of the degenerate solitons remain unchanged. When the non-degenerate solitons interact with the bound-state solitons, amplitudes and widths of the bound-state solitons remain unchanged.

Tropical limit of matrix solitons and entwining Yang\u2013Baxter maps

We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

Quasi-Grammian solutions of a multi-component short pulse equation

In this paper, we study the standard binary Darboux transformation of the multi-component short pulse (SP) equation. For this purpose, we construct the Darboux matrix from the particular eigenvector solutions to the Lax pair not only in the direct space but also in the adjoint space and obtain the binary Darboux matrix. By iterating its standard binary Darboux transformation, we obtain quasi-Grammian soliton solutions for the multi-component SP equation. We also derive explicitly, loop-soliton solutions for the one and two component cases.

General soliton solutions to a reverse‐time nonlocal nonlinear Schrödinger equation

AbstractGeneral soliton solutions to a reverse‐time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one‐soliton solutions and two‐soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev–Petviashvili equation

For a variable coefficient Kadomtsev–Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painlevé analysis. The N-fold binary Darboux transformation is presented in a compact form. As an application, the multi-lump, higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained. Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.

Lax pair, binary Darboux transformation and dark solitons for the three-component Gross–Pitaevskii system in the spinor Bose–Einstein condensate

In this paper, we investigate the dark solitons for the three-component Gross–Pitaevskii system, which describes the $$F=1$$ spinor Bose–Einstein condensate, with F denoting the atom’s spin. We construct a Lax pair and a binary Darboux transformation for such a system. Based on a nonzero seed solution, we obtain the kink and dark solitons corresponding to the ferromagnetic and polar states, respectively. Moreover, we obtain the W-shaped dark soliton. Interactions between the two dark solitons are discussed. We find that the interaction is inelastic if a kink soliton is included.

Solitons, breathers and rogue waves for the three-component Gross–Pitaevskii equations in the spinor Bose–Einstein condensates

In this paper, we construct a binary Darboux transformation for the three-component Gross–Pitaevskii equations, which describe the hyperfine spin F=1 spinor Bose–Einstein condensates. Single-, double-hump bright solitons and bright-two soliton are obtained via the binary Darboux transformation with a zero seed solution, in which the bright-two soliton describes interaction between two solitons with the same spectral variable. Kink-type, double-hump, double kink-type breathers, two-breather and kink-type two-breather are derived with a nonzero seed solution, in which the latter two kinks breathers describing the interactions between two breathers with the same spectral variable have not been reported before, to our knowledge. Based on the breather solution, we obtain the rogue-wave solution. Beak-type rogue wave is presented.

Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg‐de Vries equations

This paper considers the coupled complex modified Korteweg‐de Vries (mKdV) equations and presents a binary Darboux transformation for the equations. As a direct application, we give a classification of general soliton solutions derived from vanishing and non‐vanishing backgrounds, on the basis of the dynamical behavior of the solutions. Special types of solutions in the presented solutions include breathers, bright‐bright solitons, bright‐dark solitons, bright‐W‐shaped solitons, and rogue wave solutions. Furthermore, dynamics and interactions of vector bright solitons are exhibited.

Binary Darboux transformation of time-discrete generalized lattice Heisenberg magnet model

Abstract The standard binary Darboux transformation is used to find the solutions of time-discrete generalized lattice Heisenberg magnet (GLHM) model. We compute quasi-Grammian multisoliton solutions by iterating the binary Darboux transformation and the explicit soliton solutions for the SU(2) case are also obtained.

Binary Darboux transformation and vector-soliton-pair interactions with the negatively coherent coupling in a weakly birefringent fiber

Soliton-based data transmission provides the hope of surmounting the limitations caused by the dispersion of optical fibers. In this letter, we investigate the coherently coupled nonlinear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber. With respect to the two interacting modes in the fiber, we construct a binary Darboux transformation with the zero seed solutions, which is different from the existing one, and analytically and graphically study the vector bright solitons and their interactions. Besides the vector degenerate and nondegenerate solitons, the vector soliton pairs are derived with the associated parameter conditions. Two interaction mechanisms between the two degenerate solitons, with the phase shifts or not, are used to explain the elastic and inelastic interactions between the soliton pairs and degenerate solitons. From the elastic interactions between the nondegenerate solitons and soliton pairs, we find that the amplitudes of the formers are changed and converted back to the initial shapes in two components. Elastic interactions between the two soliton pairs are exhibited as well.

Quasi-Grammian solutions of the generalized Heisenberg magnet model

In this paper we use standard binary Darboux transformation to obtain the quasi-Grammian multi-soliton solutions of generalized Heisenberg magnet model in two dimensions. We also discuss the model based on the Lie group SU(n) and obtain explicit solutions of the model for the SU(2) case.

Dark solitons for a variable-coefficient AB system in the geophysical fluids or nonlinear optics

Under investigations in this paper is a variable-coefficient AB system, which describes the marginally-unstable baroclinic wave packets in a geophysical fluid or ultra-short pulses in nonlinear optics with certain inhomogeneous medium existing, while the binary Darboux transformation constructed and dark solitons derived. Moreover, effects of the coefficients $ \alpha$ (t) and $ \beta$ (t) , which are, respectively, the parameters measuring the state of the basic flow and reflecting the interaction of the wave packet and mean flow under inhomogeneous situations, are discussed in figures. Compared with the constant-coefficient AB system, there may appear some new interesting phenomena in the variable-coefficient one, for example, we find that the variable coefficients cannot only influence the amplitudes of the solitons but the velocities.

Binary Darboux Transformations of The Supersymmetric Heisenberg Magnet Model

We investigate the standard binary Darboux transformation of the supersymmetric Heisenberg model and calculate multisoliton solutions of the quasi determinants of the supersolitons of the Heisenberg magnet model by iterating the Darboux binary transformation.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

Breather and lump solutions for nonlocal Davey–Stewartson II equation

A binary Darboux transformation for the nonlocal Davey–Stewartson II equation is constructed by the Darboux matrix method. As applications, N-soliton, breather and lump solutions of the considered equation have been obtained. In particular, the dynamics of the single soliton, two-soliton, breather and lump solutions are discussed and illustrated through some figures.

Matrix Boussinesq solitons and their tropical limit

We study soliton solutions of matrix ‘good’ Boussinesq equations, generated via a binary Darboux transformation. Essential features of these solutions are revealed via their ‘tropical limit’, as also exploited in previous work about the KP equation. This limit associates a point particle interaction picture with a soliton (wave) solution.

Dark solitons for a variable-coefficient Kundu–Eckhaus equation in an inhomogeneous optical fiber with the relevant Lax pair and binary Darboux transformations

In this paper, we study a Kundu–Eckhaus equation with variable coefficients, which describes the ultra-short optical pulses in an inhomogeneous optical fiber. We construct the Lax pair under certain variable-coefficient constraints. With the gauge transformation, one/N-fold binary Darboux transformations and limit forms of the one-fold binary Darboux transformation are obtained. Based on such transformations, one/N-dark (N = 2,3, [Formula: see text]) soliton solutions under those constraints are derived. Linear, periodic and parabolic dark solitons are presented, and numerical simulations are used to investigate the influence of the group velocity dispersion on the structures of the one-dark solitons. Based on the two-dark soliton solutions under certain variable-coefficient constraints, we also discuss the influence of the group velocity dispersion on the structures of the two-dark solitons. Head-on and overtaking collisions between the two linear, parabolic and cubic-type dark solitons are presented.

Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a higher-order nonlinear Schrödinger system for the simultaneous propagation of two ultrashort optical pulses in an optical fiber. Based on the Lax pair, with respect to the two-component electric fields, infinitely-many conservation laws and one/N-fold binary Darboux transformations are derived, where N=2,3,…. Solitons are discussed: (1) Nondegenerate one dark–dark soliton, which is black or gray in each component, is obtained. ϵ has no relation with the soliton amplitude in each component, but has a linear correlation with the soliton velocities, where ϵ, σ1 and σ2 are the constant coefficients in the system; (2) Overtaking and head-on interactions between the two dark–dark solitons as well as the bound state are depicted. With the decreasing value of σ1, the gray solitons’ amplitudes increase, but the black solitons’ amplitudes decrease. With the decreasing value of σ2, the gray solitons’ amplitudes increase, but the black solitons’ amplitudes do not change; (3) Interaction among the three overtaking solitons and interaction between a bound state and one soliton are displayed. We find that the interactions between the two solitons and among the three solitons are elastic.